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Global attractivity and uniform persistence in Nicholson’s blowflies. (English) Zbl 0869.34056
Summary: The delay differential equation $\dot N(t)= -\delta N(t)+ PN(t-\tau)e^{-aN(t-\tau)},\quad t\geq 0\tag{$$*$$}$ was used by W. S. C. Gurney, S. P. Blythe and R. M. Nisbet [Nature 287, 17-21 (1980)] in describing the dynamics of Nicholson’s blowflies. We show that for $$P\leq\delta$$, every nonnegative solution of $$(*)$$ tends to zero as $$t\to\infty$$. On the other hand, for $$P>\delta$$, $$(*)$$ is uniformly persistent. Moreover, if in addition, $$(e^{\delta r}- 1)\ln{P\over\delta}<1$$, then every positive solution of $$(*)$$ tends to $$N^*={1\over a}\ln {P\over \delta}$$ as $$t\to\infty$$.

##### MSC:
 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 92D25 Population dynamics (general) 92D40 Ecology